142 XIII INTEGRATION (13.9.18) Let fi be a bounded positive measure on X. Then for each & > 0 there exists a compact subset KofX such that jj.(X - K) ^ e. This is a particular case of (13.7.9). (1 3.9.1 9) Suppose that \JL is bounded. Let (/,) be a uniformly bounded sequence of measurable functions (i.e., sup ||/J < + oo) such that limfn(x) =/(*) exists n H-+OO for almost all x e X. Then f is integrable, and we have (13.9.19.1) |/^=lim \fnd/i.. J »•+» J This is a particular case of (13.8.4). Example (13.9.20) Let JJL be a positive measure on X and let Y be a closed subspace of X. It follows from the Tietze-Urysohn theorem (4.5.1 ) that every function /e Jf K(Y) is the restriction to Y of a function g e <%£ (X) such that ||#|| = ||/||. The function /Y = g<py , which is equal to/in Y and zero outside Y, is there- fore ju-integrable (13.9.13), and |^(/Y)| g ||/|| • XSupp(/)). Write ,UY(/) = K/Y) (which clearly depends only on/); then the discussion above shows that /-» juy(/) is a positive measure on Y, which is said to be the measure induced by IL on Y. From (12.7.8), the remarks above and (13.5.7), it follows im- mediately that if /belongs to ,/(Y) and if as above we denote by/Y the func- tion which is equal to/in Y and is zero in X — Y, then we have ju*(/) = /^*(/Y). Hence we conclude (13.5.5) that this formula is valid for any mapping /: Y -+ R (with/Y defined as before). Hence, finally, a function /: Y -+ E is /iY-5ntegrable if and only if /Y is ^-integrable, and that in this case we have (13.9.20.1) f - f J ^Y r Likewise, a mapping u of Y into a topological space Z is /*Y-;measurable if and only if the mapping u': X -> Z which extends u to X by giving u'(x) an (arbitrary) constant value for x e X - Y, is /i-measurable. Conversely, consider a closed subset Y of X and a positive measure v on Y, and let \JL be the canonical extension of v to X (13.1.7). Clearly we have v = uy. Also, if/e >(Y), then the function g which is equal to/on Y and to +00 on X - Y belongs to J^(X), and we have v*(/) = ^(g\ It follows Ln is continuous.